Effect of the suction /
injection on unsteady MHD oscillatory flow through a vertical channel filled
with porous medium
M. Chitra1*, M. Suhasini2
1Associate Professor, Department of
mathematics, Thiruvalluvar University, Serkkadu, Vellore.
2Research Scholar, Department of mathematics,
Thiruvalluvar University, Serkkadu, Vellore.
*Corresponding
Author E-mail: chitratvu@gmail.com
ABSTRACT:
In this paper, we investigate the combined
effect of a transverse Magnetic field and radiative heat transfer on unsteady
oscillatory flow through a vertical channel filled with saturated porous medium
and non-uniform wall temperature with the effect of suction or injection. The
governing equation of oscillatory flow were non-dimensionalized, simplified and
solved analytically. The effect of thermal radiation and the Magnetic field
parameters on velocity profile, temperature, wall shear stress and the rate of
heat transfer under the influence of Hartmann’s number, Grashof number, Darcy
parameter, suction /injection parameter are computed graphically.
KEYWORDS: MHD oscillatory flow, Porous medium, Heat transfer,
Thermal Radiation, slip-condition, suction/injection.
INTRODUCTION:
The unsteady convection motion of an oscillatory flow
in the presence of heat source through a porous medium has been a subject of
interest of many researchers because of its application in geophysics, solid
mechanics, ground water, hydrology, oil recovery, thermal insulation, engineering
and heat storage. The study of electrically conductive fluid has many application
in Engineering problems such as heat transfer in MHD flow of viscous fluids,
nuclear reactors, geothermal energy, extraction and Boundary layer in the field
of Aerodynamics. The application of free convection and Heat transfer flow
through porous medium under the influence of magnetic field has attracted many
researchers in this field. Raptis et al [1985] studied the unsteady free
convective through a porous medium bounded by an infinite vertical plate.
O.D. Makinde and P.Y. Mhone [2005] studied the Heat
transfer to MHD oscillatory flow in a channel filled with porous medium. Mustafa et al [2008] investigated unsteady
MHD memory flow with oscillatory suction, variable free stream and heat
sources. S.N. Sahoo [2013] studied the Heat and mass transfer effect on MHD
flow of a viscoelastic fluid through a porous medium bounded by an oscillating
porous plate in slip-flow regime. Rita Choundhury et al [2011] studied the heat
transfer to MHD oscillatory viscoelastic flow in a channel filled with porous
medium.
The phenomenon of slip-flow regime has been studied by
many researchers due to its wide ranging applications in the modern science,
technology and vast ranging industrialization. M.M. Hamza et al [2011] studied
the unsteady Heat transfer to MHD oscillatory flow through a porous medium
under slip condition. K.D. Singh [2013] studied the effect of slip condition on
viscoelastic MHD oscillatory forced convection flow in a vertical channel with
heat radiation. There have been several studies on the convection heat transfer
through porous channel due to several other important suction/injection
controlled applications. Umavathi et al [2009] studied the unsteady flow of
viscous fluid through a horizontal composite channel whose half-width is filled
with porous medium. Ajibade and Jha [2010] presented the effect of suction and
injection on hydrodynamics of oscillatory fluid through parallel plates.
K.M.Joseph et al [2015] studied the chemically reacting fluid on unsteady MHD
oscillatory slip flow in a planer channel with varying temperature and
concentration in the presence of suction/injection. More recently J.A. Falade
et al [2016] studied the effect of suction or injection on the MHD oscillatory
flow through a porous channel saturated with porous medium.
In this paper, we studied the effect of suction or
injection on the slip flow of unsteady oscillatory hydromagnetic field and
radiative heat transfer through symmetric vertical channel filled with
saturated porous medium. The effect of thermal radiation and the Magnetic field
parameters on velocity profile, temperature, wall shear stress and the rate of
heat transfer under the influence of Hartmann’s number, Grashof number, Darcy
parameter, suction /injection parameters are computed graphically.
MATHEMATICAL FORMULATION:
Consider the unsteady laminar slip-flow of an
incompressible viscous electrically conducting heat generating oscillatory
fluid flow in a symmetrical vertical channel filled with the saturated porous
medium under the influence of an externally applied magnetic field and
radiative heat transfer. This external magnetic field is applied across the
normal to the channel. The fluid has small electrical conductivity and the
electromagnetic force produced is also very small. The flow is subjected to
suction at the cold wall and injection at the heated wall. We consider a
coordinate system (x*, y*) where x* lies
along the centre of the channel and y* is the distance measured in
the normal section such that y*=a is the half width of the channel.
Fig- 1: Geometry of the problem
Under the usual Boussinesq approximation the flow is
governed by the following equations:
The appropriate boundary conditions of the problem are
Where t* is the time, u* is the
axial velocity, 
is the constant horizontal velocity ,
ρ is the fluid density, 
is the kinematic viscosity, P*
is the fluid pressure, g is the gravitational acceleration , K is the porous
permeability,
σe is the electrical conductivity, B0
is the magnetic field intensity, β is the volumetric expansion, Cp
is the specific heat at constant pressure, α is the thermal radiation, kf
is the thermal conductivity, T* is the fluid temperature and
T0 is the referenced fluid temperature.
Introducing the following non-dimensional quantities
, 
, 
, 
, 
, 
, 
, 
, 
(6)
In view of the above dimensionless variables, the
basic equations (1) & (2) can be expressed in the following non-dimensional
form
Where Da is the Darcy parameter, s is the suction or
injection parameter, H is the Hartmann’s number,
Gr is the Grashof number, Pr is the prandtl
number, 
is the thermal radiation parameter .
Now the appropriate boundary conditions are
Where 
is the slip parameter.
METHOD OF SOLUTION :
In order to solve the equations (7) & (8) with
respect to the boundary conditions (9) &(10) for oscillatory flow,
let us take
(12)
(13)
Where A is any positive constant and ω is the
frequency of oscillation.
Substituting the Equations (11)-(13) in Equations (7)
& (8), we obtain
The corresponding boundary conditions can be written
as
The exact solution of the Equation (15) becomes
The rate of heat transfer is given by
The exact solution of (16) is given by
)
The shear stress is given by
)
(21)
The wall shear stress 
is given by
RESULTS AND DISCUSSIONS:
In this paper, we studied the combined effect of a
transverse Magnetic field and radiative heat transfer on unsteady oscillatory
flow through a vertical channel filled with saturated porous medium and
non-uniform wall temperature with the effect of suction or injection. The fluid
temperature, fluid velocity, rate of heat transfer and the wall shear stress
are shown graphically for different values of frequency of oscillation (ω)
, thermal radiation (), Hartmann’s number(H) ,Darcy parameter
(Da), Grashof number (Gr), and suction or injection parameter(s).
Figure (2), shows the fluid temperature for different
values of frequency of oscillation (ω). It is clear from the figure that
an increase in the frequency of oscillation decreases the fluid temperature.
This is reason to a reduction in the heat transfer as the heating frequency.
Figure (3), shows the fluid temperature for different values of thermal
radiation (
. It is clear from the figure that an
increase in the thermal radiation parameter increases the fluid temperature.
This is because of heat transfer from the heated wall to the fluid since the
fluid absorbs its own radiation. Figure (4), shows the fluid temperature for
different values of suction or injection parameter (s). It is observed from the
figure that an increase in the suction or injection parameter(s) increases the
fluid temperature. This result shows that the concavity with increase in the
suction or injection parameter is because of the heat flow from the heated
plate towards the cold plate. Figure (5), shows the rate of heat transfer (Nu)
for different values of suction or injection parameter(s). It is clear from the
figure that an increase in the suction or injection parameter decreases the
rate of heat transfer in the heated wall and increase in the region close to
the cold plate. This is due to the transfer of heat from the heated plate to
the fluid and from the fluid to the cold plate.
Figure (6), shows the fluid velocity for different
values of Darcy parameter (Da). It is clear from the figure that an in the
Darcy parameter decreases the fluid velocity. Figure (7), shows the fluid
velocity for different values of Hartmann’s number (H). It is clear from the
figure that an increase in the Hartmann’s number increases the fluid velocity. It
is observed that the maximum flow occurs in the presence of magnetic field. Figure
(8), shows the fluid velocity for different values of Grash of number (Gr). It
is clear from the figure that an increase in the Grash of number decreases the
fluid velocity. Figure (9), shows the fluid velocity for different values of
thermal radiation (. It is clear from the figure that an increase
in the thermal radiation parameter increases the fluid velocity. This is due
to the heat gained from the heated wall energized the fluid particles and the
internal heat generation enhances the fluid flow. Finally, Figure (10), shows
the wall shear stress for different values of suction or injection parameter
(s). It is clear from the figure that an increase in the suction or injection
parameter increases the wall shear stress on both the heated wall and the cold
wall.
Fig-2: Variation of fluid temperature (
with y for different values of oscillation
(ω) for fixed Pr=1, s=1, t=0,
=1.
Fig -3: Variation of fluid temperature (
with y for different values of thermal
radiation for fixed Pr=1, s=1, t=0,
Fig -4: Variation of fluid temperature ( for different values of suction or
injection parameter (s) for fixed Pr=1, ,
t=0,=1.
Fig -5:
Variation of rate of heat transfer (Nu) for different values of suction or
injection parameter(s) for fixed Pr=1, ,
t=0,=1.
Fig-6: Variation
of fluid velocity (
) for different values of Darcy number (Da) for fixed Pr=1, H=1, 
=0.1, t=0, s=1
=1,
=
.
Fig- 7:
Variation of fluid velocity () with y for different values of Hartmann’s
number (H) for fixed
Pr=1, Da=1, =0.1, t=0, s=1=1,
=.
Fig-8: Variation of fluid velocity () with y for different values of Grashof
number (Gr) for fixed
Pr=1, H=1, =0.1, t=0, s=1=1,
Fig-9: Variation
of fluid velocity () with y for different values of thermal
Radiation (
) for fixed Pr=1, H=1, =0.1, t=0, s=1,Da=1,,=.
Fig-10:
Variation of Wall shear stress with thermal radiation (
for different values of suction or injection parameter
(s) for fixed Pr=1, H=1,
=0.1, t=0, s=1=1,=.
CONCLUSION:
In this paper, the combined effect of a transverse
Magnetic field and radiative heat transfer on unsteady oscillatory flow through
a vertical channel filled with saturated porous medium and non-uniform wall
temperature with the effect of suction or injection is studied.
1. The fluid temperature (
increases significantly with the increase
of suction or injection parameter(s) and thermal radiation (
while decreases with the increase
frequency of oscillation (ω).
2. The rate of heat transfer decreases in the fluid
layer closed to the heated wall while it increases in the fluid layer closed to
the cold wall with the increase of suction or injection parameter(s).
3. The fluid velocity (u) increases significantly with
the increase of Hartmann’s number (H) and thermal radiation While decreases with the increase of Darcy
parameter and Grashof number.
4. The wall shear stress increases on both the heated
wall and cold wall with the increase of suction or injection parameter.
APPENDIX A:
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